A tag aware widget is one whose content can be controlled via a tag in a blog post.  You can read more about what you can do with tag aware widgets here.

How to Use A Tag Aware Widget

When you are subscribing to a widget, a tag icon is displayed next to any field that is Tag Aware:
http://forums.widgetbox.com/content/images/tag-aware-screenshot-1.png

When you click the tag icon, you see this:
http://forums.widgetbox.com/content/images/tag-aware-screenshot-2.png

This is what the options do:

If you check just ‘Enable Tag-Awareness’
http://forums.widgetbox.com/content/images/tag-aware-enable.png
This allows you to use the <a rel="wbx-keywords> tag in your blog posts.  For example, you might make a tag to the word "tech" by blogging this:
<a rel="wbx-keywords" title="tech" />
This sends the keyword "tech" to any field of any widget to which you just checked "Enable Tag-Awareness".  You can see an example in a blog post here.

If you additionally click ‘Support "rel" tags’:
http://forums.widgetbox.com/content/images/tag-aware-rel.png
Technorati and other tagging sites use the rel-tag specification to tag content in a blog.  We let these industry standard tags control widgets, too.  For example, you can tag the word "tech" by blogging this:
I like <a href="http://technorati.com/tag/tech" rel="tag">Technology</a>.
This sends the keyword "tech" to any field of any widget to which you checked ‘Support "rel" tags’.

If you additionally click ‘Named tag’:
http://forums.widgetbox.com/content/images/tag-aware-namedtag.png
This lets you send information to a specific parameter of a specific widget:
<a rel="wbx-keywords" name="srchTrms" title="tech" />
This will send the keyword "tech" to any widget that has a configuration parameter with a named tag "srchTrms".